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1996 | Buch

Elements of the Probability Theory Kapitel lesen Erstes Kapitel lesen

Limit Theorems for the Riemann Zeta-Function

verfasst von: Antanas Laurinčikas

Verlag: Springer Netherlands

Buchreihe : Mathematics and Its Applications

Enthalten in: Professional Book Archive

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Über dieses Buch

The subject of this book is probabilistic number theory. In a wide sense probabilistic number theory is part of the analytic number theory, where the methods and ideas of probability theory are used to study the distribution of values of arithmetic objects. This is usually complicated, as it is difficult to say anything about their concrete values. This is why the following problem is usually investigated: given some set, how often do values of an arithmetic object get into this set? It turns out that this frequency follows strict mathematical laws. Here we discover an analogy with quantum mechanics where it is impossible to describe the chaotic behaviour of one particle, but that large numbers of particles obey statistical laws. The objects of investigation of this book are Dirichlet series, and, as the title shows, the main attention is devoted to the Riemann zeta-function. In studying the distribution of values of Dirichlet series the weak convergence of probability measures on different spaces (one of the principle asymptotic probability theory methods) is used. The application of this method was launched by H. Bohr in the third decade of this century and it was implemented in his works together with B. Jessen. Further development of this idea was made in the papers of B. Jessen and A. Wintner, V. Borchsenius and B.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Elements of the Probability Theory
Abstract
This chapter is motivated by needs of following parts of the book. It contains notions and results of the theory of probability which are used in studying of the distribution of values of some functions defined by the Dirichlet series. Most of the material consists of well-known facts, and their proofs can be found in monographs on the theory of probability.
Antanas Laurinčikas
Chapter 2. Dirichlet Series and Dirichlet Polynomials
Abstract
In this chapter we define the object of the investigation in our book: the Dirichlet series, the Riemann zeta-function and the Dirichlet L-functions. We also give some classical results concerning the behaviour of these series.
Antanas Laurinčikas
Chapter 3. Limit Theorems for the Modulus of the Riemann Zeta-Function
Abstract
In this chapter the limit theorems on the modulus of the function ζ(s) in the half-plane σ ≥ 1/2 will be proved. The attention will be devoted mainly to the cases when σ → 1/2+0 or σ = 1/2.
Antanas Laurinčikas
Chapter 4. Limit Theorems for the Riemann Zeta-Function on the Complex Plane
Abstract
Since the function ζ(s) is complex-valued the distribution of its values is reflected more adequately by limit theorems on the complex plane C. In this chapter we will obtain such theorems on the half-plane σ ≥ 1/2.
Antanas Laurinčikas
Chapter 5. Limit Theorems for the Riemann Zeta-Function in the Space of Analytic Functions
Abstract
Let D 1 = {s ∈ C: 1/2 < σ < 1} and D 2 = {s ∈ C: σ > 1}. We define the probability measures
$${P_{j,T}}(A) = V_T^\tau (\zeta (s + i\tau ) \in A)$$
on (H(D j ), Β(H(D j ))), j = 1, 2. The aim of this chapter is to prove that the measures P j,T , converge weakly to some measure as T → ∞. Let D = {s ∈ C: σ > 1/2}.
Antanas Laurinčikas
Chapter 6. Universality Theorem for the Riemann Zeta-Function
Abstract
In this chapter we apply the limit theorem for the Riemann zeta-function in the space H(D 1) to obtain one of magnificent properties of this function — the universality property. Roughly speaking, this property asserts that any analytic function can be approximated uniformly on compact subsets of D l by translations of ζ(s).
Antanas Laurinčikas
Chapter 7. Limit Theorem for the Riemann Zeta-Function in the Space of Continuous Functions
Abstract
Let C ∞ = C ∪ {∞} be the Riemann sphere and let d(s 1, s 2) be a metric on C ∞ given by the formulae
$$d({s_1},{s_2}) = \frac{{2\left| {{s_1} - {s_2}} \right|}}{{\sqrt {1 + {{\left| {{s_1}} \right|}^2}} \sqrt {1 + {{\left| {{s_2}} \right|}^2}} }},d(s,\infty ) = \frac{2}{{\sqrt {1 + {{\left| {{s_2}} \right|}^2}} }},d(\infty ,\infty ) = 0.$$
Antanas Laurinčikas
Chapter 8. Limit Theorems for Dirichlet L-Functions
Abstract
The properties of Dirichlet L-functions with a fixed modulus are similar to those of the Riemann zeta-function. Therefore all limit theorems proved in the previous chapter can be stated also for Dirichlet L-functions. In this chapter we will give only few limit theorems which describe the asymptotic behaviour of the Dirichlet L-functions.
Antanas Laurinčikas
Chapter 9. Limit Theorem for the Dirichlet Series with Multiplicative Coefficients
Abstract
Let g(m) be a complex-valued multiplicative function such that |g(m)| ≤ 1, and let
$$Z(s) = \sum\limits_{m = 1}^\infty{\frac{{9(m)}}{{{m^2}}}} $$
(0.1)
for σ > 1. The asymptotic mean value of the function g(m), i.e.
$$\frac{1}{x}\sum\limits_{m \leqslant x} {g(m)} ,x \to \infty ,$$
(0.2)
is of great importance in the analytic number theory. The best results of such kind are obtained using the method of generating Dirichlet series, see, for example, (Halász, 1968; Levin and Timofeev, 1971; Kubilius, 1962, 1974; Elliott, 1979; Tenenbaum, 1992). This method is based on the representation of the mean value (0.2) by the contour integral of the function Z(s). For the evaluation of this integral it is necessary to know the behaviour of the function Z(s) in the neighbourhood of the line σ = 1. Thus from the functional properties of the function Z(s) the asymptotics of the mean value of its coefficients g(m) follows. It appears that an inverse relation exists, too: the functional properties of Z(s) depend on the asymptotic behaviour of the mean value (0.2). In this chapter the asymptotics of the mean value of the coefficients of the Dirichlet series are used to prove a limit theorem for the function Z(s) in the space of analytic functions. From this theorem the universality and the functional independence of Z(s) follow.
Antanas Laurinčikas
Backmatter
Metadaten
Titel
Limit Theorems for the Riemann Zeta-Function
verfasst von
Antanas Laurinčikas
Copyright-Jahr
1996
Verlag
Springer Netherlands
Electronic ISBN
978-94-017-2091-5
Print ISBN
978-90-481-4647-5
DOI
https://doi.org/10.1007/978-94-017-2091-5